International Journal of

ADVANCED AND APPLIED SCIENCES

EISSN: 2313-3724, Print ISSN: 2313-626X

Frequency: 12
line decor
  
line decor

 Volume 10, Issue 10 (October 2023), Pages: 78-85

----------------------------------------------

 Technical Note

Exact solutions of classes of second order nonlinear partial differential equations reducible to first order

 Author(s): 

 Noureddine Mhadhbi 1, Sameh Gana 2, *, Hamad Khalid Alharbi 3, 4

 Affiliation(s):

 1Department of Mathematics, College of Sciences and Arts, King Abdulaziz University, P.O. Box 344, Rabigh Campus, 21911, Saudi Arabia
 2Department of Basic Sciences, Deanship of Preparatory Year and Supporting Studies, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam, 34212, Saudi Arabia
 3Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia
 4Applied College, Imam Mohammad Ibn Saud Islamic University, Riyadh, Saudi Arabia

  Full Text - PDF

 * Corresponding Author. 

  Corresponding author's ORCID profile: https://orcid.org/0000-0003-2083-8583

 Digital Object Identifier: 

 https://doi.org/10.21833/ijaas.2023.10.009

 Abstract:

This paper illustrates the successful implementation of the method of variation of parameters in combination with the method of characteristics and other techniques to obtain exact solutions for a wide range of partial differential equations. The proposed approach reduces partial differential equations (PDEs) to first-order differential equations, referred to as classical equations, including Bernoulli, Ricatti, and Abel equations. In addition, the techniques proposed have the ability to produce precise solutions for nonlinear second order PDEs. For each PDE class, the method's effectiveness is demonstrated through illustrative examples.

 © 2023 The Authors. Published by IASE.

 This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

 Keywords: Partial differential equations, Non-linear partial differential, equations, Variation of parameters, Method of characteristics

 Article History: Received 4 May 2023, Received in revised form 8 September 2023, Accepted 20 September 2023

 Acknowledgment 

No Acknowledgment.

 Compliance with ethical standards

 Conflict of interest: The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

 Citation:

 Mhadhbi N, Gana S, and Alharbi HK (2023). Exact solutions of classes of second order nonlinear partial differential equations reducible to first order. International Journal of Advanced and Applied Sciences, 10(10): 78-85

 Permanent Link to this page

 Figures

 Fig. 1 Fig. 2 Fig. 3 Fig. 4 Fig. 5 Fig. 6 Fig. 7 

 Tables

 No Table

----------------------------------------------   

 References (25)

  1. Ablowitz MA and Clarkson PA (1991). Solitons, nonlinear evolution equations and inverse scattering transform. Cambridge University Press, Cambridge, UK. https://doi.org/10.1017/CBO9780511623998   [Google Scholar] PMid:9905818
  2. Dubey VP, Kumar R, Kumar D, Khan I, and Singh J (2020). An efficient computational scheme for nonlinear time fractional systems of partial differential equations arising in physical sciences. Advances in Difference Equations, 2020: 46. https://doi.org/10.1186/s13662-020-2505-6   [Google Scholar]
  3. Fan E (2000). Extended tanh-function method and its applications to nonlinear equations. Physics Letters A, 277(4-5): 212-218. https://doi.org/10.1016/S0375-9601(00)00725-8   [Google Scholar]
  4. Guo Y, Cao X, Liu B, and Gao M (2020). Solving partial differential equations using deep learning and physical constraints. Applied Sciences, 10(17): 5917. https://doi.org/10.3390/app10175917   [Google Scholar]
  5. Hassan M, Abdel-Razek M, and Shoreh H (2014). New exact solutions of some (2+1)-dimensional nonlinear evolution equations via extended Kudryashov method. Reports on Mathematical Physics, 74(3): 347-358. https://doi.org/10.1016/S0034-4877(15)60006-4   [Google Scholar]
  6. Higgins BG (2019). Introduction to the method of characteristics. University of California, Davis, USA.   [Google Scholar]
  7. Hounkonnou MN and Sielenou PA (2009). Classes of second order nonlinear differential equations reducible to first order ones by variation of parameters. ArXiv Preprint ArXiv: 0902.4175. https://doi.org/10.48550/arXiv.0902.4175   [Google Scholar]
  8. Kamke E (1977). Differentialgleichungen: Lösungsmethoden und Lösungen I. Gewöhnliche Differentialgleichungen. B.G. Teubner Publishers, Stuttgart, Germany. https://doi.org/10.1007/978-3-663-05925-7   [Google Scholar]
  9. Kečkić Jovan D (1976). Additions to Kamke's treatise: Variation of parameters for nonlinear second order differential equations. Publikacije Elektrotehničkog Fakulteta. Serija Matematika i Fizika, 544/576: 31-36.   [Google Scholar]
  10. Kevorkian J (1990). Partial differential equations: Analytical solutions techniques. Springer, Texts, USA. https://doi.org/10.1007/978-1-4684-9022-0   [Google Scholar]
  11. Kudryashov NA and Loguinova NB (2008). Extended simplest equation method for nonlinear differential equations. Applied Mathematics and Computation, 205(1): 396-402. https://doi.org/10.1016/j.amc.2008.08.019   [Google Scholar]
  12. Lu D, Hong B, and Tian L (2006). Bäcklund transformation and n-soliton-like solutions to the combined KdVBurgers equation with variable coefficients. International Journal of Nonlinear Science 2(1): 3-10.   [Google Scholar]
  13. Murphy GM (1960). Ordinary differential equations and their solutions. Generic New York, USA.   [Google Scholar]
  14. Myint U and Debnath L (2007). Linear partial differential equations for scientists and engineers. Springer Science & Business Media, Berlin, Germany.   [Google Scholar]
  15. Olver PJ (2014). Introduction to partial differential equations. Springer, Berlin, Germany. https://doi.org/10.1007/978-3-319-02099-0   [Google Scholar]
  16. Panayotounakos DE and Zarmpoutis TI (2011). Construction of exact parametric or closed form solutions of some unsolvable classes of nonlinear odes (Abel's nonlinear ODEs of the first kind and relative degenerate equations). International Journal of Mathematics and Mathematical Sciences, 2011: 387429. https://doi.org/10.1155/2011/387429   [Google Scholar]
  17. Polyanin AD and Manzhirov AV (2006). Handbook of mathematics for engineers and scientists. CRC Press, Boca Raton, USA.   [Google Scholar]
  18. Rhee HK, Aris R, and Amundson NR (1986). First-order partial differential equations. Prentice Hall, Hoboken, USA.   [Google Scholar]
  19. Shikuo L, Zuntao F, Shida L, and Qiang Z (2001).  Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Physics Letters A, 289(1-2): 69-74. https://doi.org/10.1016/S0375-9601(01)00580-1   [Google Scholar]
  20. Wang M, Li X, and Zhang J (2008). The (G′/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Physics Letters A, 372(4): 417-423. https://doi.org/10.1016/j.physleta.2007.07.051   [Google Scholar]
  21. Wazwaz AM (2004). A sine-cosine method for handling nonlinear wave equations. Mathematical and Computer Modelling, 40(5-6): 499-508. https://doi.org/10.1016/j.mcm.2003.12.010   [Google Scholar]
  22. Zachmanoglou EC and Thoe DW. (1986). Introduction to partial differential equations with applications. Courier Corporation, Chelmsford, USA.   [Google Scholar]
  23. Zhang D (2005b). Doubly periodic solutions of the modified Kawahara equation. Chaos, Solitons and Fractals, 25(5): 1155-1160. https://doi.org/10.1016/j.chaos.2004.11.084   [Google Scholar]
  24. Zhang H (2005a). New exact travelling wave solutions for some nonlinear evolution equations. Chaos, Solitons and Fractals, 26(3): 921-925. https://doi.org/10.1016/j.chaos.2005.01.035   [Google Scholar]
  25. Zwillinger D (1998). Handbook of differential equations. Academic Press, Cambridge, USA.   [Google Scholar]